Abstract

Kodaira fibred surfaces are a remarkable example of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications. Our tour through algebraic surfaces and their moduli (with results valid also for higher dimensional varieties) shall deal with fibrations, questions on monodromy and factorizations in the mapping class group, old and new results on Variation of Hodge Structures, Galois coverings, deformations and rigid varieties (there are rigid Kodaira fibrations). These questions lead to interesting algebraic surfaces, for instance surfaces isogenous to a product with automorphisms acting trivially on cohomology, hypersurfaces in Bagnera-de Franchis varieties, Inoue-type surfaces, remarkable surfaces constructed from VHS.